4.1.4 Expected Shortfall (CVaR)

Basic Prerequisites: 03 · Coherent Risk Measures and Probability & Measure Theory.

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1. Intuition & Practical Objective

Expected Shortfall (ES) - also called Conditional VaR, CVaR, Mean Shortfall, or Tail-VaR - is the average loss on the days VaR is breached. It is the direct answer to VaR’s most damning silence: “and then what?”

This page gives the three equivalent definitions of ES (tail mean, quantile integral, Rockafellar–Uryasev minimum), proves the two non-obvious facts - that the quantile integral equals the tail mean, and that ES $\ge$ VaR always - and verifies all three numerically. The practical payoff: ES is coherent and convex-optimisable, which is why it, not VaR, is the modern default (and the Basel FRTB standard).

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2. Mathematical Ground Truth & Derivations

2.1 The three definitions

For a loss $L$ and confidence level $\alpha \in (0,1)$:

$$\text{(Tail mean)}{\mathrm{ES}}_{\alpha }(L)=\mathbb{E}[L\mid L\ge {\mathrm{VaR}}_{\alpha }(L)].$$ $$\text{(Quantile integral)}{\mathrm{ES}}_{\alpha }(L)=\frac{1}{1-\alpha }{\int }_{\alpha }^1{\mathrm{VaR}}_u(L)du.$$ $$\text{(Rockafellar–Uryasev)}{\mathrm{ES}}_{\alpha }(L)=\underset{\beta \in \mathbb{R}}{\min }\{\beta +\frac{1}{1-\alpha }\mathbb{E}[(L-\beta {)}^{+}]\},(t{)}^{+}=\max (t,0).$$

The tail mean integrates the loss over the worst $(1-\alpha )$ fraction of outcomes - the region beyond VaR. The quantile integral averages VaR across confidence levels $u\in (\alpha ,1]$; for continuous distributions these coincide. For discrete distributions the tail mean needs a correction, and the quantile integral is the clean definition:

$${\mathrm{ES}}_{\alpha }(L)=\frac{1}{1-\alpha }\left(\mathbb{E}\right[L{1}_{\{L\ge {\mathrm{VaR}}_{\alpha }\}}]-{\mathrm{VaR}}_{\alpha }(\mathbb{P}(L\ge {\mathrm{VaR}}_{\alpha })-(1-\alpha )\left)\right).$$

2.2 The Rockafellar–Uryasev characterization (2000, Thm. 1)

Define the convex function $F_{\alpha }(\beta )=\beta +\frac{1}{1-\alpha }\mathbb{E}[(L-\beta {)}^{+}]$. Its subgradient is $1-\frac{1}{1-\alpha }\mathbb{P}(L\ge \beta )$. Theorem 1: $F_{\alpha }$ is convex, continuously differentiable, minimised at $\beta ={\mathrm{VaR}}_{\alpha }$, and its minimum equals ${\mathrm{ES}}_{\alpha }$:

$$\boxed{{\mathrm{ES}}_{\alpha }=\underset{\beta }{\min }{F}_{\alpha }(\beta ),{\mathrm{VaR}}_{\alpha }=\text{left endpoint of }\arg \underset{\beta }{\min }{F}_{\alpha }(\beta )}$$

Consequences: (i) you can compute ES without first computing VaR; (ii) minimising ES over a portfolio is a convex program (often a plain LP after sampling: $F_{\alpha }\approx \beta +\frac{1}{q(1-\alpha )}{\sum }_k(L_k-\beta {)}^{+}$ with auxiliary $u_k\ge 0$), while VaR minimisation may have multiple local extrema (02 · §2.2).

2.3 ES $\ge$ VaR, always

Since $F_{\alpha }(\beta )=\beta +(1-\alpha {)}^{-1}\mathbb{E}[(L-\beta {)}^{+}]$ and $\mathbb{E}[L\mid L\ge \mathrm{VaR}]\ge \mathrm{VaR}$ on the tail, the minimum $F_{\alpha }(\mathrm{VaR})={\mathrm{ES}}_{\alpha }$ cannot fall below ${\mathrm{VaR}}_{\alpha }$. Rockafellar–Uryasev: “the $\alpha$-VaR is never more than the $\alpha$-CVaR” - portfolios with low CVaR necessarily have low VaR as well.

2.4 Closed form and the normal benchmark

For $L\sim \mathcal{N}(\mu ,{\sigma }^2)$ (Hull eq. 22.1):

$${\mathrm{VaR}}_{\alpha }=\mu +\sigma z_{\alpha },{\mathrm{ES}}_{\alpha }=\mu +\sigma \frac{\phi (z_{\alpha })}{1-\alpha },z_{\alpha }={\Phi }^{-1}(\alpha ).$$

At $\alpha =0.975$: $\mathrm{ES}=2.337803$ vs $z_{0.99}=2.326348$ - the reason FRTB’s $97.5\%$ ES matches the old $99\%$ VaR in the normal benchmark. Under fat tails (Student-$t$, EVT), the same quantile gives a much larger ES (EVT §ES).

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3. Computational Implementation - three representations, verified to agree

Monte Carlo normal sample; verify the empirical tail mean equals the quantile integral equals the analytic formula. Stdlib only.

All three agree to Monte Carlo precision (the small residual is $O(1/\sqrt{N})$ and the discrete-quantile granularity of the tail mean). ES exceeds VaR by the expected $14.6\%$ normal gap.

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4. Failure Modes & First-Principles Breakdowns

1. ES is harder to estimate. It averages a handful of tail points, so its sampling variance exceeds VaR’s - a $99\%$ ES from a 250-day window is noisier than the VaR it sits above (05 · §3). Coherence buys aggregation safety at the cost of estimation stability; use filtered/EVT methods (McNeil–Frey) for tail efficiency.
2. Coherence does not mean elicitable. Gneiting (2011) showed ES is not elicitable on its own (VaR is). This is the honest counter-argument to ES and the reason FRTB backtests ES indirectly, through its VaR component and joint VaR–ES scoring. Coherence and backtestability are different desiderata.
3. Definition drift on discrete books. For loss pmfs with atoms, “mean of losses $\ge$ VaR” over-counts the boundary atom; use the corrected quantile-integral form (§2.1) so ES equals ${\lim }_{{\alpha }^{\prime }\downarrow \alpha }$ of the smooth tail mean. Two desks can otherwise disagree.
4. Horizon scaling is only valid i.i.d. ${\mathrm{ES}}_N={\mathrm{ES}}_1\sqrt{N}$ (Hull) presumes i.i.d. returns; with autocorrelation or volatility clustering the true multi-day ES differs - McNeil–Frey show multi-day conditional ES beats naive $\sqrt{N}$ scaling.
5. ES is still a single number. It summarises the whole tail by its mean; a strategy can shape the distribution beyond ES (a very rare but catastrophic jump) to evade it - hence the parallel need for stress testing (Stress Testing).

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5. References

• Rockafellar, R.T. & Uryasev, S., Optimization of Conditional Value-at-Risk, Journal of Risk 2(3):21–41 (2000)
• Artzner, Delbaen, Eber & Heath, Coherent Measures of Risk (1999)
• Acerbi, C. & Tasche, D., On the Coherence of Expected Shortfall, J. Banking & Finance 26(7):1487–1503 (2002)
• Hull, Options, Futures, and Other Derivatives
• McNeil & Frey, Estimation of Tail-Related Risk Measures…, J. Empirical Finance 7 (2000)
• Gneiting, T., Making and Evaluating Point Forecasts, JASA 106(494) (2011)

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6. Connected Graph Bridges

• Back: 03 · Coherent Risk Measures · Index Hub
• Forward: 05 · Failure Modes & Practice · Euler & Basel ES
• Sibling: Extreme Value Theory & Fat Tails · Estimation Methods